The Cottagers entered the tie needing to turn things around after suffering a 1-0 defeat in the first-leg last Friday.
Fulham enjoyed a good start this week, with Abdoul Camara forcing goalkeeper Scott Carson into a fine save after just nine minutes.
The hosts went close to breaking the deadlock on 25 minutes, but Aleksandar Mitrovic could only send his header over the crossbar.
Derby, meanwhile, took a while to get going and had to wait until the 38th minute for a chance, although Andre Wisdom fired horribly wide of the mark.
Carson came to the rescue for Derby just before the half-time break as he produced a fine save to keep out Mitrovic's header from point-blank range.
Two minutes into the second half, Fulham took the lead on the night, with Ryan Sessegnon smashing the ball home from close range to make it 1-1 on aggregate.
The home side then took the lead after 66 minutes when Denis Odoi sent a fine header past Carson following some good work from Kamara down the right.
The final 20 minutes saw the home side sit deep as they looked to hold on to their two-goal advantage on the night.
Matej Vydra nearly pulled a goal back for Derby in second-half stoppage time, but he dragged his effort wide of the mark as Fulham held on for the win.
The London outfit will now face the winner of the other semi-final second-leg clash between Middlesbrough and Aston Villa, the latter of which holds a 1-0 advantage.